Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Summation condition number — where it appears

The total size of a sum's terms divided by the size of its result, Σ|xᵢ| ÷ |Σxᵢ|. It is a property of the data alone, and it decides both how much rounding a summation exposes and how far two machines' answers for it can be from each other.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

110¹10⁻¹¹10⁻⁸10⁻⁵pieces the vector was divided intodistance from the exact sum, relativethe published boundκ · uone vector, one algorithmdistinct answers21runs26spread, in ulps2.3·10⁷κ of the sum10⁸bound ÷ worst error2.6·10⁴nobody chose pand no answer is the answer

The same program, twice

One vector of 4,096 numbers, one summation algorithm, one precision, twenty-six runs — and twenty-one different answers. Nothing in the program chose between them, every one of them satisfies the textbook bound, and the exactly rounded answer is not among them.

machine · reduction order 10²10⁴10⁶10⁸10¹⁰10¹²10⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹κ of the sumrelative sizethe boundκ · umeasured spreadtwo curves and one constantspread ÷ κu, low0.25spread ÷ κu, high0.26bound ÷ spread, low7932decades swept10the spread is computablethe bound cannot see the order

A bound every answer satisfies

The classical bound on a summation error is correct, it covers all twenty-six answers one vector produced, and it is 7,932 times larger than the difference between them. A statement true of every ordering cannot say which ordering you got.

machine · reduction order 051210241536204810⁻¹⁷10⁻¹⁴terms consumederror accumulated so farone accumulator8 piecesthe steps are the partial sumspeak partial sum32the answer1.6·10⁻⁵error, one piece7.6·10⁻¹⁴error, 8 pieces2.4·10⁻¹⁴mean at p = 648.1·10⁻¹⁵the walk sets the sizeand nothing sets the value

Where the disagreement comes from

The error of a reduction is a walk whose step length is the spacing of the running total, not of the answer. That one sentence predicts the size of the disagreement to a factor of two, explains why dividing the work makes it smaller, and explains why the value cannot be predicted at all.

machine · reduction order κ of the suma component of b − Ax1.01·10¹⁷ad − bc, near-degenerate3.6·10¹⁶qᵢᵀqⱼ, an orthogonality check7.39·10¹⁵pᵀAp, a curvature409zᵀAz, a trace probe41.8rᵀr, a residual norm1measured, not assumedhighest10¹⁷lowest1above 10¹⁰3terms64sums of squares are safeand nobody decides anything from one

The vector that hides it

Every quick demonstration of a parallel sum uses positive numbers, and positive numbers are the one family where the effect is absent. Measured on six inner products this site already computes, the summation condition number runs from exactly 1 to 10¹⁷ — and the safe end is where nobody makes a decision.

machine · summation 110¹-101pieces the inner product was summed inqᵢᵀqⱼ ÷ 2.78·10⁻¹⁷the true valuethe norm and its partsdistinct values5of runs10pairs with no fixed sign125of pairs1128κ of this inner product1.1·10¹⁷‖QᵀQ − I‖ moves by1the aggregate is stableand no entry of it is

An inner product with no fixed sign

‖QᵀQ − I‖ is how this site turns "orthogonal" into a number, and across ten partitionings it moves by 2.4%. The entries it is built from are not so lucky: 125 of the 1,128 off-diagonal pairs take both signs, and one of them takes five different values including zero.

machine · orthogonality κ of the suma component of b − Ax8.09·10¹⁷ad − bc, near-degenerate3.6·10¹⁶qᵢᵀqⱼ, an orthogonality check3.26·10¹⁶zᵀAz, a trace probe95.7pᵀAp, a curvature73.7rᵀr, a residual norm1measured, not assumedhighest8.1·10¹⁷lowest1above 10¹⁰3terms128sums of squares are safeand nobody decides anything from one

Two machines, one certificate

Nothing a solver returns says which of its answers you got. Four things could be reported instead — the summation condition number, the partition count, an exactly accumulated residual and a directed-rounding interval — and each costs about one pass over data the routine already has in hand.

machine · regression tolerance

Named alongside it

The objects these essays reach for when they reach for this one.

Reduction orderRun-to-run variationError accumulationUnit roundoffBitwise reproducibilityOrthogonality lossReduction treeSummationA-priori boundAlgorithm selectionAssociativityCancellation

All concepts